Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 001, 19 pages      arXiv:1301.0180
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

Changzheng Qu a, Junfeng Song b and Ruoxia Yao c
a) Center for Nonlinear Studies, Ningbo University, Ningbo, 315211, P.R. China
b) College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, P.R. China
c) School of Computer Science, Shaanxi Normal University, Xi'an, 710062, P.R. China

Received September 28, 2012, in final form December 27, 2012; Published online January 02, 2013

In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex Camassa-Holm equation and the multi-component modified Camassa-Holm equation are provided. It is shown that these equations arise from non-streching invariant curve flows respectively in the three-dimensional Euclidean geometry, the two-dimensional Möbius sphere and n-dimensional sphere Sn(1). Integrability to these systems is also studied.

Key words: invariant curve flow; integrable system; Euclidean geometry; Möbius sphere; dual Schrödinger equation; multi-component modified Camassa-Holm equation.

pdf (445 kb)   tex (27 kb)


  1. Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge, 1991.
  2. Anco S.C., Bi-Hamiltonian operators, integrable flows of curves using moving frames and geometric map equations, J. Phys. A: Math. Gen. 39 (2006), 2043-2072, nlin.SI/0512051.
  3. Anco S.C., Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces, J. Geom. Phys. 58 (2008), 1-37, nlin.SI/0703041.
  4. Anco S.C., Hamiltonian flows of curves in G/SO(N) and vector soliton equations of mKdV and sine-Gordon type, SIGMA 2 (2006), 044, 17 pages, nlin.SI/0512046.
  5. Anco S.C., Asadi E., Quaternionic soliton equations from Hamiltonian curve flows in HPn, J. Phys. A: Math. Theor. 42 (2009), 485201, 25 pages, arXiv:0905.4215.
  6. Asadi E., Sanders J.A., Integrable systems in symplectic geometry, Glasg. Math. J. 51 (2009), 5-23.
  7. Calini A., Ivey T., Marí-Beffa G., Remarks on KdV-type flows on star-shaped curves, Phys. D 238 (2009), 788-797, arXiv:0808.3593.
  8. Camassa R., Holm D.D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664, patt-sol/9305002.
  9. Chou K.S., Qu C.Z., Integrable equations arising from motions of plane curves, Phys. D 162 (2002), 9-33.
  10. Chou K.S., Qu C.Z., Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci. 13 (2003), 487-517.
  11. Chou K.S., Qu C.Z., Integrable motions of space curves in affine geometry, Chaos Solitons Fractals 14 (2002), 29-44.
  12. Chou K.S., Qu C.Z., Motions of curves in similarity geometries and Burgers-mKdV hierarchies, Chaos Solitons Fractals 19 (2004), 47-53.
  13. Constantin A., Kolev B., Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv. 78 (2003), 787-804, math-ph/0305013.
  14. Constantin A., Kolev B., Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlinear Sci. 16 (2006), 109-122, arXiv:0911.5058.
  15. Doliwa A., Santini P.M., An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A 185 (1994), 373-384.
  16. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  17. Fokas A.S., On a class of physically important integrable equations, Phys. D 87 (1995), 145-150.
  18. Fuchssteiner B., Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D 95 (1996), 229-243.
  19. Fuchssteiner B., Fokas A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981), 47-66.
  20. Geng X.G., Xue B., A three-component generalization of Camassa-Holm equation with N-peakon solutions, Adv. Math. 226 (2011), 827-839.
  21. Goldstein R.E., Petrich D.M., The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991), 3203-3206.
  22. Guha P., Olver P.J., Geodesic flow and two (super) component analog of the Camassa-Holm equation, SIGMA 2 (2006), 054, 9 pages, nlin.SI/0605041.
  23. Gui G., Liu Y., Olver P.J., Qu C.Z., Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., to appear.
  24. Hasimoto H., A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-485.
  25. Hone A.N.W., Wang J.P., Integrable peakon equations with cubic nonlinearity, J. Phys. A: Math. Theor. 41 (2008), 372002, 10 pages, arXiv:0805.4310.
  26. Ivey T.A., Integrable geometric evolution equations for curves, in The Geometrical Study of Differential Equations (Washington, DC, 2000), Contemp. Math., Vol. 285, Amer. Math. Soc., Providence, RI, 2001, 71-84.
  27. Kamran N., Olver P., Tenenblat K., Local symplectic invariants for curves, Commun. Contemp. Math. 11 (2009), 165-183.
  28. Kouranbaeva S., The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys. 40 (1999), 857-868, math-ph/9807021.
  29. Langer J., Perline R., Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), 71-93.
  30. Li Y.Y., Qu C.Z., Shu S.C., Integrable motions of curves in projective geometries, J. Geom. Phys. 60 (2010), 972-985.
  31. Marí Beffa G., Bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds, Pacific J. Math. 247 (2010), 163-188.
  32. Marí Beffa G., Conformal analogue of the Adler-Gel'fand-Dikii bracket in two dimensions, J. Phys. A: Math. Gen. 33 (2000), 4689-4707.
  33. Marí Beffa G., Geometric realizations of bi-Hamiltonian completely integrable systems, SIGMA 4 (2008), 034, 23 pages, arXiv:0803.3866.
  34. Marí Beffa G., Hamiltonian evolution of curves in classical affine geometries, Phys. D 238 (2009), 100-115.
  35. Marí Beffa G., On completely integrable geometric evolutions of curves of Lagrangian planes, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 111-131.
  36. Marí Beffa G., Poisson brackets associated to the conformal geometry of curves, Trans. Amer. Math. Soc. 357 (2005), 2799-2827.
  37. Marí Beffa G., Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces, Proc. Amer. Math. Soc. 134 (2006), 779-791.
  38. Marí Beffa G., Olver P.J., Poisson structures for geometric curve flows in semi-simple homogeneous spaces, Regul. Chaotic Dyn. 15 (2010), 532-550.
  39. Marí Beffa G., Sanders J.A., Wang J.P., Integrable systems in three-dimensional Riemannian geometry, J. Nonlinear Sci. 12 (2002), 143-167.
  40. Misiolek G., A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24 (1998), 203-208.
  41. Musso E., Motions of curves in the projective plane inducing the Kaup-Kupershmidt hierarchy, SIGMA 8 (2012), 030, 20 pages, arXiv:1205.5329.
  42. Nakayama K., Segur H., Wadati M., Integrability and the motion of curves, Phys. Rev. Lett. 69 (1992), 2603-2606.
  43. Novikov V., Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor. 42 (2009), 342002, 14 pages.
  44. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, 2nd ed., Springer-Verlag, New York, 1993.
  45. Olver P.J., Invariant submanifold flows, J. Phys. A: Math. Theor. 41 (2008), 344017, 22 pages.
  46. Olver P.J., Rosenau P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (1996), 1900-1906.
  47. Olver P.J., Sokolov V.V., Integrable evolution equations on associative algebras, Comm. Math. Phys. 193 (1998), 245-268.
  48. Pinkall U., Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), 328-332.
  49. Qiao Z., A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys. 47 (2006), 112701, 9 pages.
  50. Sanders J.A., Wang J.P., Integrable systems in n-dimensional conformal geometry, J. Difference Equ. Appl. 12 (2006), 983-995.
  51. Sanders J.A., Wang J.P., Integrable systems in n-dimensional Riemannian geometry, Mosc. Math. J. 3 (2003), 1369-1393, math.AP/0301212.
  52. Schäfer T., Wayne C.E., Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D 196 (2004), 90-105.
  53. Sharpe R.W., Differential geometry. Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, Vol. 166, Springer-Verlag, New York, 1997.
  54. Song J.F., Qu C.Z., Integrable systems and invariant curve flows in centro-equiaffine symplectic geometry, Phys. D 241 (2012), 393-402.
  55. Song J.F., Qu C.Z., Qiao Z.J., A new integrable two-component system with cubic nonlinearity, J. Math. Phys. 52 (2011), 013503, 9 pages.
  56. Tao T., Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, Vol. 106, Amer. Math. Soc., Providence, RI, 2006.
  57. Terng C.L., Thorbergsson G., Completely integrable curve flows on adjoint orbits, Results Math. 40 (2001), 286-309.
  58. Valiquette F., Geometric affine symplectic curve flows in R4, Differential Geom. Appl. 30 (2012), 631-641.
  59. Wang J.P., Generalized Hasimoto transformation and vector sine-Gordon equation, in SPT 2002: Symmetry and Perturbation Theory (Cala Gonone), World Sci. Publ., River Edge, NJ, 2002, 276-283.
  60. Wo W., Qu C.Z., Integrable motions of curves in S1×R, J. Geom. Phys. 57 (2007), 1733-1755.

Next article   Contents of Volume 9 (2013)